Theory of probability and Mathematical statistics
$D$- and $A$-optimal designs of experiments for trigonometric regression with heteroscedastic observations
V. P. Kirlitsa Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus
Abstract:
Herein for the regression function $y(x)=\theta_{1}+\displaystyle\sum_{s=1}^{k}(\theta_{2s}\cos{sx}+\theta_{2s+1} \sin{sx})$, representing a trigonometrical sum of an
$k$, order, we constructed continuous
$D$- and
$A$-optimal designs of experiments
$\varepsilon_{n}^{0}= \begin{Bmatrix} x_{1}^{0},\dots, x_{n}^{0}\\ \frac{1}{n},\dots, \frac{1}{n} \end{Bmatrix}$ with points of a spectrum $x_{i}^{0}=\frac{2\pi(i-1)}{n}+ \varphi, i=\overline{1,n}, n\geq 2k+1$, where
$\varphi$ -is an arbitrary angle
$(\varphi\geq 0)$ for which the determinant of the information matrix of the experiment design is not equal to zero. These designs of experiments are constructed for heteroscedastic observations with variances $\mathrm d (x)\geq \sigma^{2}, \mathrm d (x_{i}^{0})= \sigma^{2}, \sigma\neq 0,i=\overline{1,n}$. For a special case of the considered regression function
$(k=1)$ we constructed the saturated designs of experiments for
observations with unequal accuracy and dispersions accepting various values in the points of a spectrum of such plans.
Keywords:
continuous $D$- and $A$-optimal designs of experiments; trigonometric regression; homoscedastic observations; heteroscedastic observations.
UDC:
519.4
Received: 22.05.2023
Revised: 02.06.2023
Accepted: 05.06.2023
DOI:
10.33581/2520-6508-2023-2-35-44